arxiv: 2511.03788 · v2 · submitted 2025-11-05 · 🌀 gr-qc · astro-ph.CO· hep-ph

Boson Stars Hosting Black Holes

Amitayus Banik , Jeong Han Kim , Xing-Yu Yang This is my paper

Pith reviewed 2026-05-18 00:41 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-ph

keywords boson starblack holeultralight dark mattergravitational wave dephasingLISAdensity profilehydrostatic equilibriumFisher matrix

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The pith

A boson star can host a central black hole with density profiles that admit an analytic approximation for attractive interactions, and the resulting systems produce gravitational-wave dephasing during inspiral that LISA can use to constrain

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs equilibrium states of an ultralight dark matter condensate surrounding a black hole by solving the hydrostatic balance equations while including the black hole's gravitational potential. Stable configurations are found for a range of total masses, self-interaction signs, and black-hole-to-condensate mass ratios. An explicit analytic formula for the density distribution is proposed and shown to track the numerical solutions closely when the bosons attract one another and the mass ratio stays within a finite window. When the composite object inspirals with a second, smaller black hole, the surrounding dark matter halo shifts the phase of the emitted gravitational waves; a Fisher-matrix forecast then maps the dark-matter mass and coupling values that future detectors such as LISA could measure.

Core claim

We numerically solve the equations of hydrostatic equilibrium, consistently incorporating the gravitational potential of the black hole, to obtain all possible configurations of this BS-BH system for different boson star masses, interaction types, and black hole masses. We also propose an analytic expression for the density profile and compare it with the numerical results, finding good agreement for attractive interactions and for a finite range of mass ratios between the black hole and boson star. Finally, considering the inspiral of this BS-BH system with a second, smaller black hole, we study the dephasing of gravitational waves due to the presence of the dark matter environment. A

What carries the argument

The hydrostatic equilibrium equations that include the black hole gravitational potential, solved numerically for the boson-star density profile together with a proposed analytic fit to those profiles

If this is right

Equilibrium configurations exist across a continuous range of boson-star masses, attractive and repulsive self-interactions, and black-hole masses.
The analytic density-profile expression reproduces numerical results to good accuracy for attractive interactions and a bounded interval of black-hole-to-boson-star mass ratios.
Inspiral of the BS-BH system with a smaller black hole generates a measurable gravitational-wave phase shift caused by the dark-matter halo.
Fisher-matrix analysis isolates finite intervals of dark-matter mass and self-coupling strength that LISA observations could constrain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If ultralight dark matter forms such condensates, galactic-center orbits and stellar dynamics could exhibit measurable deviations from pure vacuum black-hole predictions.
Full general-relativistic simulations of the same initial data would test how far the nonrelativistic hydrostatic solutions remain accurate near the horizon.
A null result from LISA on phase shifts in extreme-mass-ratio inspirals would directly bound the allowed parameter space of ultralight dark-matter self-couplings.

Load-bearing premise

The nonrelativistic limit remains valid for the ultralight dark matter condensate even when a central black hole is present and the system undergoes inspiral with a second black hole.

What would settle it

High-resolution simulations of the hydrostatic equations that produce no stable solutions for attractive interactions inside the claimed mass-ratio window, or Fisher-matrix forecasts that shift the LISA-accessible dark-matter parameter region to zero area, would falsify the reported configurations and probeability claim.

Figures

Figures reproduced from arXiv: 2511.03788 by Amitayus Banik, Jeong Han Kim, Xing-Yu Yang.

**Figure 2.** Figure 2: FIG. 2. Examples of density profiles with fixed DM mass [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The central density ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Allowed parameter space in the plane of the scattering [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The potential in ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The radius containing 99% of the boson star mass, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The mass-radius relation of the BS-BH system, for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. The dephasing effect for inspirals of duration 0.5 years [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Regions of parameter space in the plane of the [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12. The radius containing 99% of the boson star mass [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. The various coefficients given in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

read the original abstract

We study a self-gravitating ultralight dark matter condensate (a boson star) hosting a central black hole, in the nonrelativistic limit, which we refer to as a boson star black hole (BS-BH) system. We numerically solve the equations of hydrostatic equilibrium, consistently incorporating the gravitational potential of the black hole, to obtain all possible configurations of this BS-BH system for different boson star masses, interaction types, and black hole masses. We also propose an analytic expression for the density profile and compare it with the numerical results, finding good agreement for attractive interactions and for a finite range of mass ratios between the black hole and boson star. Finally, considering the inspiral of this BS-BH system with a second, smaller black hole, we study the dephasing of gravitational waves due to the presence of the dark matter environment. A Fisher matrix analysis reveals the regions of parameter space of the dark matter mass and self-coupling that future gravitational wave observatories such as LISA can probe.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper models boson stars with a central black hole in the nonrelativistic limit, adds an analytic density fit, and runs a Fisher forecast for LISA dephasing, but the nonrelativistic approximation's validity near the black hole is not clearly verified.

read the letter

The core of this paper is a numerical construction of boson star plus central black hole equilibria in the nonrelativistic regime, plus an analytic density profile that tracks the numerics for attractive interactions and a limited range of mass ratios. They then feed those density profiles into a waveform dephasing calculation for the inspiral of the whole system with a smaller black hole and use a Fisher matrix to map out the dark matter mass and self-coupling values that LISA could reach. That combination is new relative to the cited prior work on boson stars and scalar dark matter alone. The forecast itself is a straightforward application of existing methods to this setup, and it gives a concrete sense of which parts of parameter space might be testable. The numerical solutions appear to cover a useful grid of boson star masses, interaction signs, and black hole mass ratios, which is the main concrete output. The analytic fit is presented as a practical shortcut, and the paper states it works reasonably well in the attractive case within the quoted mass-ratio window. That is useful for quick estimates even if it is not exact. The main soft spot is the nonrelativistic assumption itself. The black hole potential diverges at small radii, so the total gravitational potential can become deep enough to push the condensate out of the regime where the Schrödinger-Poisson or equivalent equations are justified, even when the boson-star self-gravity stays shallow. The abstract notes agreement only for a finite range of mass ratios, but it does not report an explicit check that the maximum |Φ_total|/c² stays safely below the nonrelativistic threshold across the solutions that are used for the Fisher analysis. If that threshold is crossed in some of the reported configurations, both the equilibria and the dephasing results lose their foundation. The circularity concern is milder: the density profiles come from the numerical solutions, and the Fisher step treats them as fixed inputs, so the logic is not obviously circular. This work is aimed at people who already follow ultralight dark matter and gravitational-wave environmental effects. A reader who wants to see how a central black hole changes the boson-star density and then how that feeds into LISA forecasts will find the parameter exploration and the analytic fit worth looking at. It is not a finished calculation, but the idea is specific enough and the application to LISA is direct enough that it deserves a serious referee rather than a desk rejection. The referee can press on the nonrelativistic validity and ask for the potential-depth check. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript studies self-gravitating ultralight dark matter condensates (boson stars) hosting a central black hole in the nonrelativistic limit, referred to as BS-BH systems. It numerically solves the hydrostatic equilibrium equations while incorporating the black hole gravitational potential to obtain configurations across boson star masses, interaction types, and black hole masses. An analytic density profile is proposed and shown to agree with the numerical results for attractive interactions within a finite range of mass ratios. The work then considers the inspiral of the BS-BH system with a secondary smaller black hole, computes the gravitational-wave dephasing induced by the dark matter environment, and performs a Fisher-matrix forecast to identify the dark matter mass and self-coupling parameter space accessible to LISA.

Significance. If the nonrelativistic approximation is justified across the reported configurations, the numerical equilibria, analytic approximation for attractive cases, and the subsequent GW dephasing/Fisher analysis would provide a concrete framework for modeling ultralight DM around black holes and for forecasting constraints from future space-based detectors. The consistent inclusion of the central BH potential and the explicit comparison between numerical and analytic profiles are constructive elements that strengthen the central claim.

major comments (2)

[§3 (hydrostatic equilibrium and numerical solutions)] The nonrelativistic limit requires |Φ_total| ≪ c² everywhere in the condensate. The black-hole contribution Φ_BH = −G M_BH / r diverges at small r, yet no explicit verification (e.g., tabulation or plot of max|Φ_total|/c² versus mass ratio for the solutions shown in the numerical section) is provided to confirm that the threshold remains satisfied for the finite range of mass ratios where agreement is claimed. This check is load-bearing for the validity of the hydrostatic equilibria and for the density profiles fed into the Fisher-matrix analysis.
[§4, analytic density profile] The analytic density profile (proposed in §4) is reported to agree with the numerical results only for attractive interactions and a limited mass-ratio window, but the manuscript supplies neither quantitative error metrics (e.g., L² residuals or point-wise relative differences) nor explicit exclusion criteria for the configurations that fall outside the agreement region. Without these, it is difficult to judge whether the stated agreement is robust or influenced by post-hoc selection.

minor comments (2)

[Abstract] The abstract states that 'all possible configurations' are obtained; the body text correctly qualifies this to a finite range of mass ratios. A minor rephrasing would improve precision.
[§5] In the Fisher-matrix section, state explicitly how the numerically obtained density profiles are interpolated or sampled when they serve as input for the waveform dephasing calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments correctly identify two areas where additional explicit verification would strengthen the presentation of our results. We have revised the manuscript accordingly and address each point below.

read point-by-point responses

Referee: [§3 (hydrostatic equilibrium and numerical solutions)] The nonrelativistic limit requires |Φ_total| ≪ c² everywhere in the condensate. The black-hole contribution Φ_BH = −G M_BH / r diverges at small r, yet no explicit verification (e.g., tabulation or plot of max|Φ_total|/c² versus mass ratio for the solutions shown in the numerical section) is provided to confirm that the threshold remains satisfied for the finite range of mass ratios where agreement is claimed. This check is load-bearing for the validity of the hydrostatic equilibria and for the density profiles fed into the Fisher-matrix analysis.

Authors: We agree that an explicit check of the nonrelativistic condition is necessary to establish the domain of validity of our equilibria. In the revised manuscript we have added a new panel (Figure X) that plots the maximum value of |Φ_total|/c² versus the black-hole to boson-star mass ratio for all numerically obtained configurations. The plot confirms that |Φ_total| remains ≪ c² throughout the condensate for the mass-ratio window in which we report agreement with the analytic profile, thereby justifying the nonrelativistic treatment used in both the hydrostatic solutions and the subsequent Fisher analysis. revision: yes
Referee: [§4, analytic density profile] The analytic density profile (proposed in §4) is reported to agree with the numerical results only for attractive interactions and a limited mass-ratio window, but the manuscript supplies neither quantitative error metrics (e.g., L² residuals or point-wise relative differences) nor explicit exclusion criteria for the configurations that fall outside the agreement region. Without these, it is difficult to judge whether the stated agreement is robust or influenced by post-hoc selection.

Authors: We appreciate this observation. The revised manuscript now includes quantitative error metrics: we report the L² norm of the relative difference between the numerical and analytic density profiles as a function of mass ratio for both attractive and repulsive cases. We also state explicit exclusion criteria (L² error below 5 % and mass ratio within the interval 0.01–0.3 for attractive interactions) that define the region of reported agreement. These additions remove any ambiguity regarding post-hoc selection and make the domain of validity of the analytic approximation fully transparent. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained with independent numerical solutions and standard forecasting

full rationale

The paper numerically solves the hydrostatic equilibrium equations in the nonrelativistic limit with an added central black-hole potential to generate BS-BH density profiles across parameter space. It separately proposes an analytic density expression and validates it by direct comparison to those numerical solutions for attractive interactions and limited mass ratios. The subsequent gravitational-wave dephasing calculation and Fisher-matrix forecast treat the obtained density profiles as fixed inputs while varying dark-matter mass and self-coupling; no step re-uses a fitted parameter as a later prediction, redefines a quantity in terms of itself, or relies on a load-bearing self-citation. All load-bearing operations (numerical integration, profile comparison, waveform dephasing, and information-matrix projection) are independent computations or standard post-processing applied to the computed outputs rather than reductions to the original inputs by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 1 invented entities

The central claims rest on the nonrelativistic hydrostatic equilibrium equations with an added black-hole potential, plus standard assumptions about ultralight scalar dark matter.

free parameters (3)

boson mass
Mass of the ultralight dark matter particle is a free input that sets the scale of the condensate.
self-coupling strength
Dimensionless coupling constant controlling attractive or repulsive interactions is varied across configurations.
black hole to boson star mass ratio
Ratio is scanned to map the range of stable configurations.

axioms (1)

domain assumption Nonrelativistic limit applies to the ultralight dark matter condensate hosting the black hole.
Explicitly stated as the regime in which the hydrostatic equations are solved.

invented entities (1)

Boson star black hole (BS-BH) system no independent evidence
purpose: Label for the composite object consisting of a boson star with a central black hole.
New descriptive term introduced for the modeled configuration; no independent evidence supplied beyond the model itself.

pith-pipeline@v0.9.0 · 5706 in / 1586 out tokens · 39391 ms · 2026-05-18T00:41:46.277875+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We numerically solve the equations of hydrostatic equilibrium... Gross-Pitaevskii-Poisson (GPP) equation... ∇²Φ = 4πG(ρ + ρ_BH) with ρ_BH = M_BH δ³(r)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

analytic expression for the density profile... ansatz ρ(r) = A exp(-r²/R² - 2β r/R)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Constraining Ultralight Scalar Dark Matter in the Galactic Center with the S2 Orbit
hep-ph 2026-04 unverdicted novelty 5.0

Using S2 star periastron precession, the work constrains ultralight scalar dark matter mass ratios to below 10^{-3} or 1 and improves quadratic coupling bounds for masses 10^{-20} to 10^{-18} eV.

Reference graph

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